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Gauss's law for magnetism, $\nabla \cdot \mathbf {B} =0$, is most directly interpreted as a sort of Kirchhoff's current law for magnetism, stating that while magnetic fields can be drawn between points (dipoles), they can't spring automagnetically (bad joke?) from single points. In other words, no monopoles.

And yet massive work is done on trying to "find the elusive magnetic monopole", notably recently at the London Centre for Nanotechnology. To ask the question, why the uncertainty? Considering the Maxwell equations and the Lorentz force law form the core of basically all of our models of electromagnetism, why the intensive search for something the mathematics say isn't there?

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    $\begingroup$ You're looking at the wrong equation. For monopoles, Maxwell's hardly relevant -- you have to use the non-abelian generalisation, to wit, the Yang-Mills equations. $\endgroup$ – AccidentalFourierTransform Jun 24 '18 at 19:23
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    $\begingroup$ Mathematics does not say monopoles don't exist. An empirical physical law says that they don't exist. If they turn out to exist that law will need to be revised. $\endgroup$ – tfb Jun 24 '18 at 19:24
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    $\begingroup$ Related: Why do physicists believe that there exist magnetic monopoles? and links therein. $\endgroup$ – Qmechanic Jun 24 '18 at 19:27
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Because Maxwell's laws are empirical. We construct them by joining up what we know (Gauss law, Faraday-Lenz's law, and so on).

Ampère's law needed to be modified in order to solve some problems (Ampère-Maxwell's law aftwerwards), and that's what makes light possible.

So why couldn't we wrong? What if someday we find something so that $\vec{\nabla}\cdot \vec{B}\neq0$ anymore? Then monopoles would exist. It wouldn't be the first time we discover new things that change everything.

In short, mathematics say there aren't monopoles, but we did build that math.

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If magnetic monopoles existed, Maxwell's equations would have to be extended and that would be a big break for the discoverer. The reward is high but the chances are low. There is as yet no indication at all that magnetic monopoles exist.

In condensed matter quasiparticles have been demonstrated that may be considered magnetic monopoles.

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